Kinetic theory of lattice waves

Transcription

1 Intro DNLS Kinetic FPU Comments Extras Kinetic theory of (lattice) waves based on joint works with Herbert Spohn (TU München), Matteo Marcozzi (U Geneva), Alessia Nota (U Bonn), Christian Mendl (Stanford U), Jianfeng Lu (Duke U) Bristol 2018

2 Intro DNLS Kinetic FPU Comments Extras Setup Evol H Init Theorem Cumulants Part I Time-correlations in stationary states of the discrete NLS [JL and H. Spohn, Invent. Math. 183 (2011) ]

3 Intro DNLS Kinetic FPU Comments Extras Setup Evol H Init Theorem Cumulants Setup for the rigorous result 3 Finite lattice: L 2, Λ = {0, 1,..., L 1} d Periodic BC: All arithmetic mod L Dual lattice: Λ = {0, 1 L,..., L 1 L }d Integration = finite sum: dk f (k) := 1 f (k) Λ Λ k Λ Dirac delta = finite sum: δ Λ (k) := Λ 1 {k mod 1 = 0} Fourier transform: (x Λ, k Λ ) f (k) = f (y)e i2πk y f (x) = dk f (k )e i2πk x y Λ Λ

4 Intro DNLS Kinetic FPU Comments Extras Setup Evol H Init Theorem Cumulants Evolution equations 4 Discrete nonlinear Schrödinger equation i d dt ψ t(x) = y Λ α(x y)ψ t (y) + λ ψ t (x) 2 ψ t (x) ψ t : Λ C, t R λ > 0 (defocusing) Harmonic coupling determined by α : Z d R. α has finite range (for instance, nearest neighbour) We assume also α( x) = α(x)

5 Intro DNLS Kinetic FPU Comments Extras Setup Evol H Init Theorem Cumulants Conservation laws 5 Hamiltonian function H Λ (ψ) = x,y Λ α(x y)ψ(x) ψ(y) λ x Λ ψ(x) 4 Relate q x, p x R to ψ by ψ(x) = 1 2 (q x + ip x ) NLS equivalent to the Hamiltonian equations q x = px H Λ, ṗ x = qx H Λ Thus H Λ (ψ t ) is conserved By explicit differentiation, also x ψ t(x) 2 is conserved

6 Intro DNLS Kinetic FPU Comments Extras Setup Evol H Init Theorem Cumulants Initial state 6 Probability distribution of ψ = ψ 0 (Grand canonical ensemble) 1 Z λ β,µ e β(h Λ(ψ) µ ψ 2 ) [d(re ψ(x)) d(im ψ(x))] x Λ Define ω : T d R by ω = F x k α. We consider only β > 0 and µ < min k ω(k) Also the Gaussian measure at λ = 0 is well-defined Z λ β,µ > 0 is the normalization constant Let E denote expectation over the initial data

7 Intro DNLS Kinetic FPU Comments Extras Setup Evol H Init Theorem Cumulants Properties of the system 7 The solution ψ t exists and is unique for all t R with any initial data ψ 0 C Λ. (conservation laws) Initial state is stationary: E[F (ψ t )] = E[F (ψ 0 )] Also invariant under periodic translations: E[F (τ x ψ)] = E[F (ψ)], (τ x ψ)(y) = ψ(y + x) Translations commute with the time-evolution: τ x ψ t = ψ t ψ0 =τ x ψ 0 Gauge invariance : similar invariance properties hold for translations of total phase, ψ 0 (x) e iϕ ψ 0 (x), ϕ R. Thus, for instance, E[ψ t ] = 0, E[ψ t ψ t ] = 0, E[ψ t (x ) ψ t (x)] = E[ψ 0 (0) ψ t t (x x )]

8 Intro DNLS Kinetic FPU Comments Extras Setup Evol H Init Theorem Cumulants Field-field correlation function 8 Fix test-functions f, g l 2, and assume they have finite support. Observable Q λ Λ (τ) := E[ f, ψ 0 e iωλ τλ 2 ĝ, ψ τλ 2 ] Under additional assumptions on the decay of equilibrium correlations and on the dispersion relation: Theorem There is τ 0 > 0 such that for all τ < τ 0 lim lim λ 0 Λ Qλ Λ (τ) = dk ĝ(k) f (k)w (k)e Γ 1 (k) τ iτγ 2 (k) T d

9 Intro DNLS Kinetic FPU Comments Extras Setup Evol H Init Theorem Cumulants Summary of the main result 9 Loosely: for all not too large t = O(λ 2 ), E[ ψ 0 (k ) ψt (k)] δ Λ (k k)w (k)e iωλ ren (k)t e λ2 t Γ 1 (k) W (k) = (β(ω(k) µ)) 1 = covariance function for λ = 0 ω λ ren(k) = ω(k) + λr 0 + λ 2 Γ 2 (k) Γ 1 (k) 0 k-space correlation decays exponentially in t, as dictated by e λ2 t Γ 1 (k). Nearest neighbour couplings (ω nn (k) = c d ν=1 cos(2πkν )) satisfy all of our assumptions if d 4

10 Intro DNLS Kinetic FPU Comments Extras Setup Evol H Init Theorem Cumulants Γ j (k) are real, and Γ(k) = Γ 1 (k) + iγ 2 (k) is given by Γ(k 1 ) = 2 0 dt dk 2 dk 3 dk 4 δ(k 1 + k 2 k 3 k 4 ) (T d ) 3 e it(ω1+ω2 ω3 ω4) (W 2 W 3 + W 2 W 4 W 3 W 4 ) with ω i = ω(k i ), W i = W (k i ). 1 Γ 1 (k 1 ) = 2π W (k 1 ) 2 dk 2 dk 3 dk 4 δ(k 1 + k 2 k 3 k 4 ) (T d ) 3 δ(ω 1 + ω 2 ω 3 ω 4 ) 4 W (k i ) 2Γ 1 (k) 0 coincides with the loss term of the linearisation of C NL around W Can be derived following the same recipe as for kinetic equations (more later... ) i=1

11 Intro DNLS Kinetic FPU Comments Extras Setup Evol H Init Theorem Cumulants Main tool to handle non-gaussian initial data 11 Moments to cumulants formula Cumulant expansion For any index set I, [ ] E ψ 0 (k i, σ i ) = i I S π(i ) A S ( ] [δ Λ k i )C A (k A, σ A ), i A where the sum runs over all partitions S of the index set I. Here truncated correlation (cumulant) functions are C n (k, σ) := 1 {x1 =0}e i2π [ n n i=1 x i k i E ψ 0 (x i, σ i ) x Λ n i=1 and for any random variables a 1,..., a n [ n ] trunc E a i := κ[a1,..., a n ] = η1 ηn ln E[e i η i a i ] i=1 ] trunc η=0

12 Intro DNLS Kinetic FPU Comments Extras Setup Evol H Init Theorem Cumulants Assumption: decay of initial correlations 12 l 1-clustering of the equilibrium measure For sufficiently small λ and for all n 4 the truncated correlation functions (cumulants) should satisfy [ n trunc sup 1 {x1 =0} E ψ 0 (x i, σ i )] λc n 0 n! Λ,σ {±1} n x Λ n i=1 For n = 2 should have E[ψ 0 (0) ψ 0 (x)] E[ψ 0 (0) ψ 0 (x)] λ=0 λ2c0 2 x L/2 L= Proven in [Abdesselam, Procacci, and Scoppola, 2009] Estimates imply that C n < cumulant expansion encodes all singularities in k i The rest is just analysis of oscillatory integrals...

13 Intro DNLS Kinetic FPU Comments Extras l p-clustering DNLS Part II Asymptotic independence, evolution of cumulants, and kinetic theory [JL and M. Marcozzi, J. Math. Phys. 57 (2016) (27pp)]

14 Intro DNLS Kinetic FPU Comments Extras l p-clustering DNLS Goal: Kinetic theory of homogeneous DNLS 14 Assume that the initial state is translation invariant, gauge invariant and with fast decay of correlations Then there always is w t (x) such that E[ψ t (x ) ψ t (x)] = w t (x x) Kinetic conjecture: W τ = lim λ 0 lim Λ (F w τλ 2) solves a homogeneous non-linear Boltzmann Peierls equation τ W τ (k) = C NL [W τ ( )], C NL [h](k 1 ) = 4π dk 2 dk 3 dk 4 δ(k 1 + k 2 k 3 k 4 ) (T d ) 3 δ(ω 1 + ω 2 ω 3 ω 4 ) [h 2 h 3 h 4 h 1 (h 2 h 3 + h 2 h 4 h 3 h 4 )]

15 Intro DNLS Kinetic FPU Comments Extras l p-clustering DNLS Why study cumulants? 15 Observation: If y, z are independent random variables we have E[y n z m ] = E[y n ]E[z m ] 0 whereas the corresponding cumulant is zero if n, m 0. Consider a random lattice field ψ(x), x Z d, which is (very) strongly mixing under lattice translations: Assume that the fields in well separated regions become asymptotically independent as the separation grows. Then κ[ψ(x), ψ(x + y 1 ),..., ψ(x + y n 1 )] 0 as y i. How fast? l 1 - or l 2 -summably? Not true for corresponding moments: E[ ψ(x) 2 ψ(x + y) 2 ]

16 Intro DNLS Kinetic FPU Comments Extras l p-clustering DNLS l p -clustering states 16 Call the field l p -clustering if sup x Z d y (Z d ) n 1 κ[ψ(x), ψ(x + y 1 ),...] p <, n. If 1 p 2, can take Fourier-transform in y functions F (n) (x, k), L in x Z d and L 2 -integrable for k (T d ) n 1. l 1 -clustering implies that F (n) (x, k) is continuous and uniformly bounded ( helps in nonlinearities) Many examples of l 1 -clustering thermal Gibbs states, e.g., discrete NLS [Abdesselam, Procacci, Scoppola]

17 Intro DNLS Kinetic FPU Comments Extras l p-clustering DNLS Time-evolution of cumulants? 17 As before, consider deterministic evolution of ψ t, with random initial data for ψ 0. Cumulants are multilinear and permutation invariant Solution? n t κ[ψ t (x l ) n l=1 ] = κ[ t ψ t (x l ), ψ t (x l ) l l] l=1 How to iterate into a closed hierarchy? Computations often simplified by using Wick polynomial representation of t ψ t

18 Intro DNLS Kinetic FPU Comments Extras l p-clustering DNLS An example: Discrete NLS 18 Consider the DNLS with initial data for ψ 0 which is l 1 -clustering Gauge invariant: ψ 0 (x) e iθ ψ 0 (x) for any θ R also ψ t will then be gauge invariant. Slowly varying in space: the cumulants vary only slowly under spatial translations Then, for instance, (x, y) E[ψ 0 (x) ψ 0 (x + y)] is slowly varying in x and l 1 -summable in y. Denote W t (x, k) = y Z d e i2πk y E[ψ t (x) ψ t (x + y)] In the spatially homogeneous case, W t (x, k) = W t (k) = Wigner function (as defined in earlier works)

19 Intro DNLS Kinetic FPU Comments Extras l p-clustering DNLS Higher order Wigner functions 19 ψ t (x, +1) = ψ t (x) and ψ t (x, 1) = ψ t (x) IF l p -clustering is preserved by the time-evolution, should study Order-n Wigner functions : x Z d, k (T d ) n 1, σ {±1} n F (n) t (x, k, σ) = y (Z d ) n 1 e i2πk y κ[ψ t (x, σ 1 ), ψ t (x + y 1, σ 2 ),..., ψ t (x + y n 1, σ n )] 1 Now W t (x, k) = F (2) t (x, k, ( 1, 1)) 2 Its derivative involves only W and F (4) 3 Compute also the derivative of F (4) and solve both by integrating out the free evolution (in Duhamel form) 4 Insert F (4) result into W, and check/argue that the remaining F (4) and F (6) can be ignored for t 1, λ 1

20 Intro DNLS Kinetic FPU Comments Extras l p-clustering DNLS With ω(k) = α(k) and ρ t (x) = E[ ψ t (x) 2 ] = dk W t (x, k), t W t (x, k) = i α(z)e i2πk z (W t (x, k) W t (x z, k)) z i2λ (ρ t (x + z) ρ t (x)) dk e i2πz (k k) W t (x, k ) z ( ) iλ dk 1dk 2 F (4) t (x, k 1, k 2, k k 1 k 2) F (4) t (x, k 1, k 2, k) 1 2π 1 kω(k) x W t (x, k) + 2λ x ρ t (x) 2π kw t (x, k) + λ 2 C NL [W t (x, )](k) O(λ) term is of Vlasov Poisson type The first two terms vanish for spatially homogeneous states Using the same recipe in Part I yields Γ(k)

21 Intro DNLS Kinetic FPU Comments Extras Pre-therm Phonons Wigner BPeq Numerical Part III Kinetic theory of the onsite FPU-chain (DNKG) with pre-thermalization [C.B. Mendl, J. Lu, and JL, Phys. Rev. E 94 (2016) (9 pp)]

22 Intro DNLS Kinetic FPU Comments Extras Pre-therm Phonons Wigner BPeq Numerical Pre-thermalization 22 Pre-thermalization = quasi-thermalization The system is thermalized but with wrong equilibrium states (e.g. extra conservation laws) Happens if there are quasi-conserved observables with very long equilibration times (e.g. with relaxation times of order e L for system size L) Problematic, since may interfere with relaxation of the true conservation laws: For instance, if diffusive, L 2 e L for system L 1

23 Intro DNLS Kinetic FPU Comments Extras Pre-therm Phonons Wigner BPeq Numerical FPU-type chains 23 We consider a chain of classical particles with nearest neighbour interactions, dynamics defined by The Hamiltonian H = N 1 j=0 [ 1 2 p2 j q2 j 1 2 δ(q j 1q j + q j q j+1 ) λq4 j λ 0 is the coupling constant for the onsite anharmonicity If λ = 0, the evolution is explicitly solvable using normal modes whose dispersion relations are ±ω(k) with ω(k) = ( 1 2δ cos(2πk) ) 1/2 The parameter 0 < δ 1 2 controls the pinning onsite potential This model is expected to have (diffusive) normal heat conduction for λ, δ > 0 ]

24 Intro DNLS Kinetic FPU Comments Extras Pre-therm Phonons Wigner BPeq Numerical Normal modes 24 From a solution (q i (t), p i (t)) define Phonon fields â t (k, σ) = 1 2ω(k) [ω(k) q(k, t) + iσ p(k, t)], σ {±1} d dt ât(k, σ) = iσω(k)â t (k, σ) iσλ σ {±1} 3 (Λ ) 3 d 3 k δ Λ ( k 3 j=1 k j ) 3 l=0 1 2ω(k l ) 3 â t (k j, σ j) j=1 Here k Λ := { N,..., N, 1 2 } for a finite periodic chain of length N (= Λ )

25 Intro DNLS Kinetic FPU Comments Extras Pre-therm Phonons Wigner BPeq Numerical Wigner function 25 Assume a spatially homogeneous state and consider the corresponding Wigner function w t (k; L) := dk â t (k ) â t (k) = e i2πy k a t (0) a t (y) Λ y Λ We expect ( kinetic conjecture ) that then the following limit exists W τ (k) = lim λ 0 lim L w λ 2 τ (k; L) Describes evolution of covariances for large lattices (L ) at kinetic time-scales (t = λ 2 τ = O(λ 2 ))

26 Intro DNLS Kinetic FPU Comments Extras Pre-therm Phonons Wigner BPeq Numerical Boltzmann Peierls equation 26 In addition, the limiting Wigner functions should satisfy the following phonon Boltzmann equation t W (k 0, t) = 12πλ 2 δ(k j=1 σ {±1} 3 ( σ j k j ) δ ω 0 + T 3 d 3 k 3 ) σ j ω j j=1 3 l=0 1 2ω l [ W 1 W 2 W 3 + W 0 (σ 1 W 2 W 3 + σ 2 W 1 W 3 + σ 3 W 1 W 2 ) ] Here W i = W (k i, t), ω i = ω(k i ) For chosen nearest neighbour interactions, the collision δ-functions have solutions only if 3 j=1 σ j = 1

27 Intro DNLS Kinetic FPU Comments Extras Pre-therm Phonons Wigner BPeq Numerical Boltzmann Peierls equation 27 Kinetic equation (spatially homogeneous initial data) t W (k 0, t) = 9π 4 λ2 d 3 1 k T ω 3 0 ω 1 ω 2 ω 3 δ(ω 0 + ω 1 ω 2 ω 3 )δ(k 0 + k 1 k 2 k 3 ) [ ] W 1 W 2 W 3 + W 0 W 2 W 3 W 0 W 1 W 3 W 0 W 1 W 2 Stationary solutions are W (k) = 1 β (ω(k) µ ) µ results from number conservation which is broken by the original evolution (then expect µ = 0)

28 Intro DNLS Kinetic FPU Comments Extras Pre-therm Phonons Wigner BPeq Numerical Spatially homogeneous kinetic theory 28 Microsystem w t (k) = dk a t (k ) a t (k) Dynamics: free evolution + λ perturbation Initial state: translation invariant & chaotic (weak coupling) τ W τ (k) = C[W τ ](k) τ S[W τ ] = σ[w τ ] Boltzmann equation for W τ = lim w λ 2 τ λ 0 S = kinetic entropy (H-function) σ = entropy production 0 σ[w eql ] = 0 W eql from an equilibrium state (classifies stationary solutions)

29 Intro DNLS Kinetic FPU Comments Extras Pre-therm Phonons Wigner BPeq Numerical Numerical simulations (Christian Mendl) 29 To compare in more detail to kinetic theory, we consider several stochastic, periodic and translation invariant initial data: Then W sim (k, t) = 1 N a(k, t) 2 Computing the covariance from simulated equilibrium states (one parameter, β) and fitting numerically to the kinetic formula (two parameters, β, µ ) yields β β µ As expected, β β and µ 0 for large β

30 Intro DNLS Kinetic FPU Comments Extras Pre-therm Phonons Wigner BPeq Numerical Set N = 64 (periodic BC), δ = 1 4 (pinning) Consider two sets of non-equilibrium initial data: A) Bimodal momentum distribution (λ = 1): Choose an initial temperature β 0 and sample positions q j from the corresponding equilibrium distribution and the momenta p j from the bimodal distribution Z 1 exp [ ( β 0 4p 4 j 1 )] 2 p2 j B) Random phase, with given initial Wigner function (λ = 1 2, 10): Take a function W 0 (k) and compute initial q j and p j from a(k) = NW 0 (k) e iϕ(k) where each ϕ(k) is i.i.d. randomly distributed, uniformly on [0, 2π]

31 Intro DNLS Kinetic FPU Comments Extras Pre-therm Phonons Wigner BPeq Numerical A) Bimodal initial data W(k,t) W(k,t) W(k,t) (a) t = 0 k (b) t = 150 k (c) t = 250 k W(k,t) W(k,t) W(k,t) (d) t = 500 k (e) t = 1000 k (f) t = 2500 k Wigner function from simulations (blue dots) vs. solving the kinetic equation (yellow triangles) starting at t = 500. (black dashed line) Kinetic equilibrium profile fitted to (f)

32 Intro DNLS Kinetic FPU Comments Extras Pre-therm Phonons Wigner BPeq Numerical B) Random phase initial data (λ = 1 2 ) W(k,t) W(k,t) W(k,t) k k k (a) t = 0 (b) t = 250 (c) t = 500 W(k,t) 0.06 W(k,t) 0.06 W(k,t) k k k (d) t = 1000 (e) t = 5000 (f) t = Wigner function from simulations (blue dots) vs. solving the kinetic equation (yellow triangles) starting at t = 500. (f) Expected equilibrium distribution (red dot-dashed line)

33 Intro DNLS Kinetic FPU Comments Extras Pre-therm Phonons Wigner BPeq Numerical Eventual relaxation towards equilibrium? (λ = 10) Δρ(t) Δe(t) t t (a) ρ sim (t) ρ sim (t max) (b) e sim (t) e sim (t max) Figure: Time evolution of the density and energy differences using λ = 10 ( kinetic time-scale (β/λ) 2 10) and longer simulation time t max = 10 6 Will this trend continue until true equilibrium values have been reached?

34 Intro DNLS Kinetic FPU Comments Extras Open Open problems 34 Pre-thermalization: What is going on here? Is it 1D effect only?... or finite size?... or just for some initial data? Inhomogeneous initial data: Does kinetic theory perform as well?... with the Vlasov Poisson term? Proofs and proper assumptions? For rigorous proofs of time-correlations: a priori estimates for propagation of clustering for equilibrium time-correlations derived in [JL, M. Marcozzi and A. Nota, arxiv: ] Anything similar for non-stationary states?

35 Intro DNLS Kinetic FPU Comments Extras Proof Wick Simulations Entropy Appendix

36 Intro DNLS Kinetic FPU Comments Extras Proof Wick Simulations Entropy Outline of the proof 36 1 Show that it is enough to prove the result assuming t > 0 2 Iterate a Duhamel formula N 0 (λ) times to expand a t into a perturbation sum (we choose N 0! λ p, for a small p) 3 There are two types of terms in the expansion: Main terms These will contain a finite monomial of a 0 whose expectation can be evaluated using the moments to cumulants formula. Error terms These will involve also a s for some s > 0. The expectation is estimated by a Schwarz bound and stationarity of the equilibrium measure The bound involves again only finite moments of a 0.

37 Intro DNLS Kinetic FPU Comments Extras Proof Wick Simulations Entropy 4 Each cumulant induces linear dependencies between the wave vectors. These can be encoded in Feynman graphs. 5 This results in a sum with roughly (N 0!) 2 non-zero terms. However, most of these vanish in the limit λ 0, due to oscillating phase factors. 6 Careful classification of graphs: we use a special resolution of the wave vector constraints which allows an estimation based on identifying, and iteratively estimating, certain graph motives. 7 Only a small fraction of the graphs (leading graphs) will remain. These consist of graphs obtained by iterative addition of one of the 20 leading motives. 8 The limit of the leading graphs is explicitly computable, and their sum yields the result in the main theorem.

38 Intro DNLS Kinetic FPU Comments Extras Proof Wick Simulations Entropy Wick polynomials 38 Generating functions g t (λ) := ln g mom,t (λ), g mom,t (λ) := E[e λ ψt ]. Then with J λ := i J λ i, y J = i J y i, Define κ[ψ t (x) J ] = J λ g t(0), E[ψ t (x) J ] = J λ g mom,t(0) G w (ψ t, λ) = eλ ψt E[e λ ψt ] t κ[ψ t (x) J ] = λ J tg t (λ) λ=0 = λ J E[λ tψ t G w (ψ t, λ)] λ=0 = E[ t ψ t (x l ) J\l λ G w(ψ t, 0)] l J J λ G w(ψ t, 0) = :ψ t (x) J : are called Wick polynomials

39 Intro DNLS Kinetic FPU Comments Extras Proof Wick Simulations Entropy WP have been mainly used for Gaussian fields. They were introduced in quantum field theory where the unperturbed measure concerns Gaussian (free) fields Gaussian case has significant simplifications: If C j j = κ[y j, y j ] denotes the covariance matrix, G w (y, λ) = exp[λ (y y ) λ Cλ/2]. Wick polynomials are Hermite polynomials The resulting orthogonality properties are used in the Wiener chaos expansion and Malliavin calculus

40 Intro DNLS Kinetic FPU Comments Extras Proof Wick Simulations Entropy Truncated moments-to-cumulants formula [ ] E y J :y J : = π P(J J) A π (κ[y A ]1 {A J} ) (1) :y J : are µ-a.s. unique polynomials of order J such that (1) holds for every J Multi-truncated moments-to-cumulants formula Suppose L 1 is given and consider a collection of L + 1 index sequences J, J l, l = 1,..., L. Then with I = J ( L l=1 J l) [ E y J L l=1 ] :y J l : = π P(I ) A π ( ) κ[ya ]1 {A Jl, l}.

41 Intro DNLS Kinetic FPU Comments Extras Proof Wick Simulations Entropy Suppose that the evolution equation of the random variables y j (t) can be written in a form t y j (t) = I M I j (t) :y(t) I : Then the cumulants satisfy t κ[y(t) I ] = Ml I (t)e[ :y(t) I : :y(t) I \l : ] l I I where the truncated moments-to-cumulants formula implies E [ :y(t) I : :y(t) I \l : ] = ( κ[y(t)a ]1 {A I, A (I \l) }) π P(I (I \l)) A π evolution hierarchy for cumulants

42 Intro DNLS Kinetic FPU Comments Extras Proof Wick Simulations Entropy The discrete NLS equation on the lattice Z d deals with functions ψ : R Z d C which satisfy i t ψ t (x) = y Z d α(x y)ψ t (y) + λ ψ t (x) 2 ψ t (x) Assuming that E[ψ t (x)] = 0 and using the WP one gets i t ψ t (x) = y Z d α(x y) :ψ t (y): +2λρ t (x) :ψ t (x): +λ :ψ t (x) ψ t (x)ψ t (x): ρ t (x) = E[ψ t (x) ψ t (x)] = E[ ψ t (x) 2 ] This splitting was called pair truncation in [JL, Spohn] (Part I)

43 Intro DNLS Kinetic FPU Comments Extras Proof Wick Simulations Entropy Molecular dynamics simulations (Kenichiro Aoki, 2006) 43 T T 2 T + T 2 N sites 1 Simulate a chain of N particles with two heat baths (Nosé-Hoover) at ends, waiting until a steady state reached 2 Measure temperature and current profiles: T i = p 2 i, J = 1 N j J j,j+1 J i,i+1 3 Fourier s Law predicts that when T 0, N J κ(t, N). T Repeat for several T, and estimate κ(t, N) from the slope. 4 Increase N to estimate κ(t ) = lim κ(t, N). N

44 Intro DNLS Kinetic FPU Comments Extras Proof Wick Simulations Entropy Comparison to kinetic prediction 44 Simulations yield good agreement with the kinetic prediction of T 2 κ(t ) 0.28 δ 3/2 for T 0, δ small (black solid line) T = 0.1 T = 0.4 T = 4

45 Intro DNLS Kinetic FPU Comments Extras Proof Wick Simulations Entropy Evolution of entropy S(t) t microscopic kinetic (a) Bimodal initial data S(t) t microscopic kinetic (b) Random phase initial data Time evolution of entropy S(t) = dk log W (k, t) (blue dots) W = Wigner function measured from simulations (orange dots) W = solution to the kinetic equation, initial data from t = 500 simulation results